/*-*- mode:c;indent-tabs-mode:nil;c-basic-offset:4;tab-width:8;coding:utf-8 -*-│
│vi: set net ft=c ts=4 sts=4 sw=4 fenc=utf-8 :vi│
╞══════════════════════════════════════════════════════════════════════════════╡
│ Copyright (c) 2008-2016 Stefan Krah. All rights reserved. │
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#include "third_party/python/Modules/_decimal/libmpdec/bits.h"
#include "third_party/python/Modules/_decimal/libmpdec/difradix2.h"
#include "third_party/python/Modules/_decimal/libmpdec/mpdecimal.h"
#include "third_party/python/Modules/_decimal/libmpdec/numbertheory.h"
#include "third_party/python/Modules/_decimal/libmpdec/sixstep.h"
#include "third_party/python/Modules/_decimal/libmpdec/transpose.h"
#include "third_party/python/Modules/_decimal/libmpdec/umodarith.h"
/* clang-format off */
asm(".ident\t\"\\n\\n\
libmpdec (BSD-2)\\n\
Copyright 2008-2016 Stefan Krah\"");
asm(".include \"libc/disclaimer.inc\"");
/*
Cache Efficient Matrix Fourier Transform
for arrays of form 2ⁿ
The Six Step Transform
══════════════════════
In libmpdec, the six-step transform is the Matrix Fourier Transform in
disguise. It is called six-step transform after a variant that appears
in [1]. The algorithm requires that the input array can be viewed as an
R×C matrix.
Algorithm six-step (forward transform)
──────────────────────────────────────
1a) Transpose the matrix.
1b) Apply a length R FNT to each row.
1c) Transpose the matrix.
2) Multiply each matrix element (addressed by j×C+m) by r**(j×m).
3) Apply a length C FNT to each row.
4) Transpose the matrix.
Note that steps 1a) - 1c) are exactly equivalent to step 1) of the Matrix
Fourier Transform. For large R, it is faster to transpose twice and do
a transform on the rows than to perform a column transpose directly.
Algorithm six-step (inverse transform)
──────────────────────────────────────
0) View the matrix as a C×R matrix.
1) Transpose the matrix, producing an R×C matrix.
2) Apply a length C FNT to each row.
3) Multiply each matrix element (addressed by i×C+n) by r**(i×n).
4a) Transpose the matrix.
4b) Apply a length R FNT to each row.
4c) Transpose the matrix.
Again, steps 4a) - 4c) are equivalent to step 4) of the Matrix Fourier
Transform.
──
[1] David H. Bailey: FFTs in External or Hierarchical Memory
http://crd.lbl.gov/~dhbailey/dhbpapers/
*/
/* forward transform with sign = -1 */
int
six_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
{
struct fnt_params *tparams;
mpd_size_t log2n, C, R;
mpd_uint_t kernel;
mpd_uint_t umod;
mpd_uint_t *x, w0, w1, wstep;
mpd_size_t i, k;
assert(ispower2(n));
assert(n >= 16);
assert(n <= MPD_MAXTRANSFORM_2N);
log2n = mpd_bsr(n);
C = ((mpd_size_t)1) << (log2n / 2); /* number of columns */
R = ((mpd_size_t)1) << (log2n - (log2n / 2)); /* number of rows */
/* Transpose the matrix. */
if (!transpose_pow2(a, R, C)) {
return 0;
}
/* Length R transform on the rows. */
if ((tparams = _mpd_init_fnt_params(R, -1, modnum)) == NULL) {
return 0;
}
for (x = a; x < a+n; x += R) {
fnt_dif2(x, R, tparams);
}
/* Transpose the matrix. */
if (!transpose_pow2(a, C, R)) {
mpd_free(tparams);
return 0;
}
/* Multiply each matrix element (addressed by i*C+k) by r**(i*k). */
SETMODULUS(modnum);
kernel = _mpd_getkernel(n, -1, modnum);
for (i = 1; i < R; i++) {
w0 = 1; /* r**(i*0): initial value for k=0 */
w1 = POWMOD(kernel, i); /* r**(i*1): initial value for k=1 */
wstep = MULMOD(w1, w1); /* r**(2*i) */
for (k = 0; k < C; k += 2) {
mpd_uint_t x0 = a[i*C+k];
mpd_uint_t x1 = a[i*C+k+1];
MULMOD2(&x0, w0, &x1, w1);
MULMOD2C(&w0, &w1, wstep); /* r**(i*(k+2)) = r**(i*k) * r**(2*i) */
a[i*C+k] = x0;
a[i*C+k+1] = x1;
}
}
/* Length C transform on the rows. */
if (C != R) {
mpd_free(tparams);
if ((tparams = _mpd_init_fnt_params(C, -1, modnum)) == NULL) {
return 0;
}
}
for (x = a; x < a+n; x += C) {
fnt_dif2(x, C, tparams);
}
mpd_free(tparams);
#if 0
/* An unordered transform is sufficient for convolution. */
/* Transpose the matrix. */
if (!transpose_pow2(a, R, C)) {
return 0;
}
#endif
return 1;
}
/* reverse transform, sign = 1 */
int
inv_six_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
{
struct fnt_params *tparams;
mpd_size_t log2n, C, R;
mpd_uint_t kernel;
mpd_uint_t umod;
mpd_uint_t *x, w0, w1, wstep;
mpd_size_t i, k;
assert(ispower2(n));
assert(n >= 16);
assert(n <= MPD_MAXTRANSFORM_2N);
log2n = mpd_bsr(n);
C = ((mpd_size_t)1) << (log2n / 2); /* number of columns */
R = ((mpd_size_t)1) << (log2n - (log2n / 2)); /* number of rows */
#if 0
/* An unordered transform is sufficient for convolution. */
/* Transpose the matrix, producing an R*C matrix. */
if (!transpose_pow2(a, C, R)) {
return 0;
}
#endif
/* Length C transform on the rows. */
if ((tparams = _mpd_init_fnt_params(C, 1, modnum)) == NULL) {
return 0;
}
for (x = a; x < a+n; x += C) {
fnt_dif2(x, C, tparams);
}
/* Multiply each matrix element (addressed by i*C+k) by r**(i*k). */
SETMODULUS(modnum);
kernel = _mpd_getkernel(n, 1, modnum);
for (i = 1; i < R; i++) {
w0 = 1;
w1 = POWMOD(kernel, i);
wstep = MULMOD(w1, w1);
for (k = 0; k < C; k += 2) {
mpd_uint_t x0 = a[i*C+k];
mpd_uint_t x1 = a[i*C+k+1];
MULMOD2(&x0, w0, &x1, w1);
MULMOD2C(&w0, &w1, wstep);
a[i*C+k] = x0;
a[i*C+k+1] = x1;
}
}
/* Transpose the matrix. */
if (!transpose_pow2(a, R, C)) {
mpd_free(tparams);
return 0;
}
/* Length R transform on the rows. */
if (R != C) {
mpd_free(tparams);
if ((tparams = _mpd_init_fnt_params(R, 1, modnum)) == NULL) {
return 0;
}
}
for (x = a; x < a+n; x += R) {
fnt_dif2(x, R, tparams);
}
mpd_free(tparams);
/* Transpose the matrix. */
if (!transpose_pow2(a, C, R)) {
return 0;
}
return 1;
}